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Stable manifolds of holomorphic hyperbolic maps. (English) Zbl 1071.32016
Let \(\{F_j\}\) be a family of holomorphic self-maps of the unit ball of \(\mathbb C^k\) which satisfies the following estimates: \(s\| z-w\| \leq \| F_j(z)-F_j(w)\| \leq r\| z-w\| \) and \(\| F_j(z)\| \leq r\) for some \(0<s\leq r<1\) and all \(z,w\) in the unit ball. A tail is an infinite sequence \((x_\ell, x_{\ell+1},\ldots)\) with the property that \(x_{j+1}=F_j(x_j)\) and \(\ell=1\) or \(x_{\ell}\) does not belong to the image of \(F_{\ell-1}\). The set \(\Omega\) of all tails can be endowed with a complex structure and it is called the abstract basin of attraction of \(\{F_j\}\).
The authors show that there exists a biholomorphism \(\Psi:\Omega\to V\) with \(V\subseteq\mathbb C^k\) being an open domain. The domain \(V\) is Stein and Runge and it is the increasing union of biholomorphic images of the ball. Moreover, the Kobayashi metric of \(V\) vanishes identically. As a consequence, they can prove the main result of the paper. Let \(F:M\to M\) be a holomorphic automorphism of a complex manifold \(M\), \(K\subset M\) an \(F\)-invariant compact hyperbolic set. Then the stable manifolds are all biholomorphic to domains in \(\mathbb C^k\) which are Fatou-Bieberbach domains or Short \(\mathbb C^k\).

MSC:
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
37F99 Dynamical systems over complex numbers
32Q35 Complex manifolds as subdomains of Euclidean space
32F45 Invariant metrics and pseudodistances in several complex variables
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References:
[1] Bieberbach L., Preuss. Akad. Wiss. Sitzungsber pp 476–
[2] Fatou G., Bull. Soc. Math. France 47 pp 161–
[3] DOI: 10.1007/s002220200220 · Zbl 1048.37047 · doi:10.1007/s002220200220
[4] Rosay J. P., Trans. Amer. Math. Soc. 310 pp 47–
[5] DOI: 10.2307/2372437 · Zbl 0080.29902 · doi:10.2307/2372437
[6] DOI: 10.1007/s002220050234 · Zbl 0932.37028 · doi:10.1007/s002220050234
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